Geoscience Reference
In-Depth Information
small values in mercury to very large values in organic fluids. We'll consider next
the behavior of the scalar spectrum beyond the inertial subrange in the limits of
very small and very large values of
ν/γ
, and then also briefly discuss the velocity
spectrum beyond the inertial subrange.
1
Here molecular diffusion begins to act on the scalar field at wavenumbers at which
the velocity field is still effectively inviscid. The rate of destruction of scalar
fluctuations by this diffusion causes the scalar variance cascade rate to decrease
with increasing
κ
, attenuating the scalar spectrum at wavenumbers in the iner-
tial subrange of the velocity field. If we assume that
Co
min
, the eddy Corrsin
number at the end of the scalar inertial subrange, equals
Re
min
, then from
(7.12)
we have
7.1.5.1 The inertial-diffusive subrange: ν/γ
r
end
(Re
min
)
3
/
4
η
oc
∼
(
scalar spectrum
).
(7.13)
Corrsin
(
1964
) assumed that in the initial part of this subrange the scalar spectrum
continues to behave as in the inertial range,
Eq. (7.9)
, but with a wavenumber-
dependent scalar cascade rate
T(κ)
:
T(κ)
−
1
/
3
κ
−
5
/
3
.
E
c
∼
(7.14)
The rate of decrease of
T(κ)
due to smoothing of the scalar field by the molecular
diffusivity is given by
dT
dκ
=−
2
γκ
2
E
c
(κ).
(7.15)
Corrsin
(
1964
) found the solution of the system
(7.14)
and
(7.15)
to be
T(κ)
=
χ
c
exp
2
β(κη
oc
)
4
/
3
.
(7.16)
E
c
behaves like the Obukhov-Corrsin form
(7.9)
at the beginning of the inertial
range, but the diminishing spectral cascade rate causes it to fall off exponentially
from that beginning at the Obukhov-Corrsin wavenumber 1
/η
oc
.
2
β(κη
oc
)
4
/
3
,
c
=
βχ
c
−
1
/
3
κ
−
5
/
3
exp
3
3
−
−
1
Here the large kinematic viscosity
ν
begins to smooth the turbulent velocity field
at scales much larger (wavenumbers much smaller) than those at which the small
molecular diffusivity
γ
impacts the scalar field. In the range between the viscous
cutoff of the velocity spectrum but well short of the diffusive cutoff of the scalar
spectrum, the scalar field is deformed by
(/ν)
1
/
2
, the mean strain rate of the
7.1.5.2 The viscous-convective subrange: ν/γ
